Polynomial, rational and analytic first integrals for a family of 3-dimensional Lotka-Volterra systems

  • Jaume Llibre
  • Clàudia Valls


We extend the study of the integrability done by Leach and Miritzis (J Nonlinear Math Phys 13:535–548, 2006) on the classical model of competition between three species studied by May and Leonard (SIAM J Appl Math 29:243–256, 1975), to all real values of the parameters. Additionally, our results provide all polynomial, rational and analytic first integrals of this extended model. We also classify all the invariant algebraic surfaces of these models.

Mathematics Subject Classification (2000)

34C05 34A34 34C14 


Polynomial integrability Rational integrability Analytic integrability Lotka-Volterra systems 


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Authors and Affiliations

  1. 1.Departament de MatemàtiquesUniversitat Autònoma de BarcelonaBellaterra, Barcelona, CataloniaSpain
  2. 2.Departamento de MatemáticaInstituto Superior TécnicoLisbonPortugal

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